Plane Mirrors — Question 6

Question 6

Two perpendicular mirrors reverse two coordinates rather than all three

Two plane mirrors occupy x=0 and y=0. An object point lies at (a,b,c), with a and b positive. Consider images from either single reflection and from reflection in both planes.

1. List the three distinct image coordinates.

2. Write the double-reflection transformation and determine whether it preserves three-dimensional handedness.

Question 6 - Solution

Two perpendicular mirrors reverse two coordinates rather than all three

(a) Method and calculation

$$\boxed{I_x=(-a,b,c),\quad I_y=(a,-b,c)}.$$$$\boxed{I_{xy}=(-a,-b,c)}\quad\text{independent of reflection order}.$$

(b) Interpretation and further calculation

$$\mathbf r_{xy}=\operatorname{diag}(-1,-1,1)\mathbf r.$$$$\boxed{\det[\operatorname{diag}(-1,-1,1)]=+1}\Rightarrow\text{handedness preserved}.$$

Physical check

One reflection reverses handedness; two perpendicular reflections are equivalent to a 180-degree rotation about their intersection line.